aa r X i v : . [ m a t h - ph ] A p r Deriving the Gross-Pitaevskii equation
Niels BenedikterInstitute of Applied Mathematics, University of BonnEndenicher Allee 60, 53115 Bonn, GermanyE-mail: [email protected] 8, 2017
Abstract
In experiments, Bose-Einstein condensates are prepared by cooling a dilute Bose gasin a trap. After the phase transition has been reached, the trap is switched off andthe evolution of the condensate observed. The evolution is macroscopically describedby the Gross-Pitaevskii equation. On the microscopic level, the dynamics of Bose gasesare described by the N -body Schr¨odinger equation. We review our article [BdS12] inwhich we construct a class of initial data in Fock space which are energetically closeto the ground state and prove that their evolution approximately follows the Gross-Pitaevskii equation. The key idea is to model two-particle correlations with a Bogoliubovtransformation. Keywords: Bose-Einstein condensate; dilute Bose gas; Gross-Pitaevskii equation; cor-relations; many-body systems; Bogoliubov transformations; squeezed coherent states.
In dilute gases of bosonic particles, at very low temperatures a phase transition occurs anda macroscopic number of particles occupies the same one-particle state. Bose and Einsteintheoretically predicted this state of matter in 1924, considering non-interacting bosons. Theexperimental confirmation took 70 years and was rewarded with a Nobel prize in 2001,and the theoretical study of interacting Bose-Einstein condensates still poses challengingproblems.A dilute gas of N bosons can be described with the Hamilton operator H UN = N X j =1 ( − ∆ j + U ( x j )) + N X i 0) and spherically symmetric; theGross-Pitaevskii scaling N V ( N. ) models rare but strong collisions. We are interested inlarge N (in experiments, N ≫ ).A central role is played by the solution f to the zero-energy scattering equation (cid:16) − ∆ + 12 V (cid:17) f = 0 , with boundary condition f ( x ) → | x | → ∞ ) . Its solution has the form f ( x ) ≃ − a / | x | for large x , where a := (8 π ) − R f V d x is thescattering length of V . By scaling, f ( N. ) solves the zero-energy scattering equation withscaled potential N V ( N. ). 1t was proven [LSY00] that the ground state energy E N of the Hamiltonian H UN satisfieslim N →∞ E N N = min ϕ ∈ L ( R ) , k ϕ k =1 E GP ( ϕ ) , (1)with the Gross-Pitaevskii energy functional E GP ( ϕ ) := Z d x (cid:0) |∇ ϕ | + U | ϕ | + 4 πa | ϕ | (cid:1) . (2)The ground state ψ gs N exhibits [LS02] complete Bose-Einstein condensation, in the sense γ (1) ψ gs N in trace norm −−−−−−−−−−→ | ϕ GP ih ϕ GP | ( N → ∞ ) , where | ϕ GP ih ϕ GP | is the projection on the minimizer ϕ GP of the Gross-Pitaevskii functional(2), and γ (1) ψ gs N is the one-particle reduced density matrix associated with ψ gs N , i. e. the trace-class operator on L ( R ) defined through the integral kernel γ (1) ψ gs N ( x ; y ) := Z d x . . . d x N ψ gs N ( x, x , . . . , x N ) ψ gs N ( y, x , . . . , x N ) . Since we generally assume k ψ k = 1, we have tr γ (1) ψ = 1.When the traps are switched off ( U = 0), the system starts to evolve, following theSchr¨odinger equation i∂ t ψ N,t = H N ψ N,t , ψ N, = ψ gs N . It was proven [ESY10, Pic10] that γ (1) ψ N,t → | ϕ t ih ϕ t | as N → ∞ (for any fixed t > ϕ t is the solution to the non-linear Gross-Pitaevskii equation (here with initial data ϕ = ϕ GP ) i∂ t ϕ t = − ∆ ϕ t + 8 πa | ϕ t | ϕ t , ϕ = ϕ. (3)In our analysis we generalize the system to Fock space, with the advantage that we canuse initial data that are superpositions of states with different numbers of particles. Weintroduce the bosonic Fock space F := L ∞ j =0 L ( R j ) and creation/annihilation operators(more precisely operator-valued distributions) a ∗ x , a x , which create/annihilate a particle at x ∈ R . They satisfy the canonical commutation relations [ a x , a ∗ y ] = δ ( x − y ), [ a ∗ x , a ∗ y ] =0 = [ a x , a y ]. We introduce the number of particles operator N = R d x a ∗ x a x and the vacuumvector Ω = (1 , , . . . ) ∈ F . On Fock space F we define the Hamiltonian H N := Z d x ∇ x a ∗ x ∇ x a x + 12 N Z d x d y N V ( N ( x − y )) a ∗ x a ∗ y a y a x . (4)The restriction of H N to L ( R N ) coincides with the Hamiltonian H N .For g ∈ L ( R ) we define the Weyl operator W ( g ) := exp (cid:18)Z d x a ∗ x g ( x ) − h . c . (cid:19) , and for integral kernels k ∈ L ( R × R ) we introduce the Bogoliubov transformation T ( k ) := exp (cid:18) Z d x d y k ( x ; y ) a ∗ x a ∗ y − h . c . (cid:19) . (5)2he Weyl operators have the important property of shifting the operators, W ∗ ( g ) a ∗ x W ( g ) = a ∗ x + ¯ g ( x ) , W ∗ ( g ) a x W ( g ) = a x + g ( x ) , (6)whereas the Bogoliubov transformation T ( k ) acts by T ∗ ( k ) a ∗ x T ( k ) = Z d y (cid:0) a ∗ y cosh( k )( y ; x ) + a y sinh( k )( y ; x ) (cid:1) . (7)Here cosh( k )( y ; x ) and sinh( k )( y ; x ) are the integral kernels defined by the power series (in k ) of the hyperbolic cosine/sine, with product in the sense of operators.We will use a Weyl operator to generate a condensate. (The coherent state W ( g )Ωdescribes a condensate with approximately k g k particles in the one-particle state g/ k g k .)A Bogoliubov transformation, on the other hand, is used to implement correlations amongthe particles. Our main result [BdS12] is the following theorem. Theorem. Let V ≥ and V ∈ L ∩ L (cid:0) R , (1 + | x | )d x (cid:1) . Let ϕ ∈ H ( R ) with k ϕ k =1 . Let k ( x ; y ) := − N (1 − f ( N ( x − y ))) ϕ ( x ) ϕ ( y ) . Let χ ∈ F , possibly depending on N but s. t. h χ, (cid:0) N + 1 + N /N + H N (cid:1) χ i is bounded uniform in N . We consider ψ N,t := e − i H N t W ( √ N ϕ ) T ( k ) χ , the solution to the Schr¨odinger equation in Fock space, i.e. i∂ t ψ N,t = H N ψ N,t . Then there exist constants C, c > s. t. tr (cid:12)(cid:12)(cid:12) γ (1) ψ N,t − | ϕ t ih ϕ t | (cid:12)(cid:12)(cid:12) ≤ C √ N exp( c exp( c | t | )) , where ϕ t solves the Gross-Pitaevskii equation (3) with initial data ϕ = ϕ . The vector χ in the initial data describes small deviations from the squeezed coherentstate W ( √ N ϕ ) T ( k )Ω. The correlation structure is inserted already in the initial data; ourproof keeps this structure static, showing its approximate stability. Our initial data arisesnaturally as an approximation to the ground state since h W ( √ N ϕ ) T ( k ) χ, (cid:0) H N + Z d x U ( x ) a ∗ x a x (cid:1) W ( √ N ϕ ) T ( k ) χ i = N E GP ( ϕ ) + O ( N / ) . In our article [BdS12] we also discuss initial data with exact number of particles. The approach is inspired by the method of coherent states [RS09], developed for studyingthe mean-field regime. However, coherent states cannot provide a good approximation inthe Gross-Pitaevskii regime because they describe completely uncorrelated states. To takeinto account the correlations we use Bogoliubov transformations.For technical reasons we will compare the solution of the many-body Schr¨odinger equationfirst to the solution of the modified Gross-Pitaevskii equation i∂ t ϕ ( N ) t = − ∆ ϕ ( N ) t + (cid:0) N f ( N. ) V ( N. ) ∗ | ϕ ( N ) t | (cid:1) ϕ ( N ) t , ϕ ( N )0 = ϕ. (8)Since N f ( N. ) V ( N. ) → πa δ , it is easy to compare ϕ ( N ) t with the solution ϕ t of the Gross-Pitaevskii equation (3). With f the solution to the zero-energy scattering equation, let k t ( x ; y ) := − N (1 − f ( N ( x − y ))) ϕ ( N ) t ( x ) ϕ ( N ) t ( y ) . 3e try to approximate the full evolution ψ N,t = e − i H N t W ( √ N ϕ ) T ( k ) χ with the (up to thesmall deviation) squeezed coherent state W ( √ N ϕ ( N ) t ) T ( k t ) χ . Thus, inspired by [RS09], weintroduce the fluctuation dynamics U N ( t ) := T ∗ ( k t ) W ∗ ( √ N ϕ ( N ) t ) e − i H N t W ( √ N ϕ ) T ( k ) . We find the estimatetr (cid:12)(cid:12)(cid:12) γ (1) ψ N,t − | ϕ ( N ) t ih ϕ ( N ) t | (cid:12)(cid:12)(cid:12) ≤ C √ N hU N ( t ) χ, N U N ( t ) χ i . (9)Hence, to show the convergence of the many-body dynamics to the Gross-Pitaevskii equation,the central task is to bound the number of fluctuations hU N ( t ) χ, N U N ( t ) χ i uniformly in N .In the next section we explain how to obtain such a bound. In this section we use the following shorthands: W t := W ( √ N ϕ ( N ) t ), T t := T ( k t ), h . i t := hU N ( t ) χ, . U N ( t ) χ i .We intend to use Gr¨onwall’s lemma. Hence we compute the derivative ∂ t hN i t = h [ i L N ( t ) , N ] i t , (10)with L N ( t ) the time-dependent generator of U N ( t ). Explicitly L N ( t ) = ( i∂ t T ∗ t ) T t + T ∗ t (cid:0) ( i∂ t W ∗ t ) W t + W ∗ t H N W t (cid:1) T t =: ( i∂ t T ∗ t ) T t + T ∗ t L (0) N ( t ) T t . The term ( i∂ t T ∗ t ) T t is harmless. Let us focus on the second term. In L (0) N ( t ) we have( i∂ t W ∗ t ) W t = −√ N Z d x a ∗ x i∂ t ϕ ( N ) t ( x ) + h . c . ( + irrelevant scalar) . For W ∗ t H N W t we use (6) and expand. We get summands which are linear in creation andannihilation operators and formally of order N / ; moreover quadratic summands of orderone, cubics of order N − / and quartics of order N − .Unlike in the mean-field regime [RS09], where the Hartree equation implies completecancellation of the linear terms in W ∗ t H N W t with ( i∂ t W ∗ t ) W t , the modified Gross-Pitaevskiiequation (8) leaves us with the linear, large remainder N / Z d x (cid:16) N V ( N. ) (1 − f ( N. )) ∗ | ϕ ( N ) t | (cid:17) ( x ) ϕ ( N ) t ( x ) a ∗ x + h . c . (11)The key observation is that by conjugating L (0) N ( t ) with T t , using (7) and expanding, weget (among many other terms) cubic terms which are not normal-ordered. Normal-orderingthem by the canonical commutation relations gives rise to a linear term which cancels (11).Similarly we get cancellations between quadratic and quartic terms of L (0) N ( t ): we conju-gate them with T t and expand the product; then normal-ordering of quartic terms gives riseto extra quadratic terms. Using the zero-energy scattering equation we now find a cancel-lation of some quadratic terms. (For identifying this cancellation think of sinh( k t )( x ; y ) as k t ( x ; y ) and of cosh( k t )( x ; y ) as δ ( x − y ).)These cancellations are crucial; they allow us to prove the operator inequality[ i L N ( t ) , N ] ≤ H N + C t (cid:0) N /N + N + 1 (cid:1) . C t varying constants which may grow exponentially in t (since we use boundsof the form k ϕ ( N ) t k H n ≤ Ce K | t | ), but are independent of N .)Next observe that L N ( t ) = H N + other terms, where ‘other terms’ can be bounded aboveand below by ε H N (any ε > 0) and the number operator. Thus H N ≤ C t (cid:0) L N ( t ) + N /N + N + 1 (cid:1) . (12)Thus we obtain [ i L N ( t ) , N ] ≤ C t (cid:0) L N ( t ) + N /N + N + 1 (cid:1) . It is possible to control hN /N i t by h ( N + 1) /N i t =0 combined with hN i t . This implies ∂ t hN i t ≤ C t hN + 1 + L N ( t ) i t + C t h ( N + 1) /N i . To close the scheme of Gr¨onwall’s lemma, we need to control the growth of hL N ( t ) i t . Similarto the above estimates we find ∂ t hL N ( t ) i t = h ˙ L N ( t ) i t ≤ C t hL N ( t ) + N + 1 i t + C t h ( N + 1) /N i . Combining the last two bounds, we obtain (for some fixed D t to be chosen later) ∂ t h D t ( N + 1) + L N ( t ) i t ≤ C t h D t ( N + 1) + L N ( t ) i t + C t h ( N + 1) /N i . Thus, Gr¨onwall’s lemma implies that for some C, c > hL N ( t ) + D t ( N + 1) i t ≤ C exp( c exp( ct )) hL N (0) + N + 1 + N /N i . By (12) there exists a C t > L N ( t ) + C t ( N /N + N ) ≥ 0. Choosing D t := C t + 1we obtain hN i t ≤ hL N ( t ) + D t ( N /N + N ) i t ≤ C exp( c exp( ct )) , which by (9) completes the proof of the main result. Acknowledgments The author would like to thank Benjamin Schlein and Gustavo de Oliveira for useful com-ments on a previous version of this review. References [BdS12] Niels Benedikter, Gustavo de Oliveira, and Benjamin Schlein. Quantitative Deriva-tion of the Gross-Pitaevskii equation, to appear in Comm. Pure Appl. Math.Preprint arXiv:1208.0373 , 2012.[ESY10] L´aszl´o Erd˝os, Benjamin Schlein, and H.-T. Yau. Derivation of the Gross-Pitaevskiiequation for the dynamics of Bose-Einstein condensate. Ann. of Math. (2) ,172(1):291–370, 2010.[LS02] Elliot H. Lieb and Robert Seiringer. Proof of Bose-Einstein condensation for dilutetrapped gases. Phys. Rev. Lett. , 88:170409, 2002.[LSY00] Elliot H. Lieb, Robert Seiringer, and Jakob Yngvason. Bosons in a trap: A rigorousderivation of the Gross-Pitaevskii energy functional. Phys. Rev. A , 61:043602, 2000.[Pic10] Peter Pickl. Derivation of the Time Dependent Gross Pitaevskii Equation withExternal Fields. Preprint arXiv:1001.4894 , 2010.[RS09] Igor Rodnianski and Benjamin Schlein. Quantum fluctuations and rate of conver-gence towards mean field dynamics.